If,$x,y and z,$ and are all different from zero and $\left|\begin{array}{l}1+x\text{ 1 1}\\ 1\text{ 1 + y 1}\\ 1\text{ 1 1 + z}\end{array}\right|=0$, then the value of $x^{-1}+y^{-1}+z^{-1}$is

we have,$\left|\begin{array}{l}1+x\text{ 1 1}\\ 1\text{ 1 + y 1}\\ 1\text{ 1 1 + z}\end{array}\right|=0$

$\text{applying }{C}_1\to C_1-C_3\text{ and }{C}_2\to C_2-C_3$

$\Rightarrow \left|\begin{array}{l}x\text{ 0 1}\\ \text{ 0 y 1}\\ \text{ -z -z 1+z}\end{array}\right|=0$$\left({\text{Expanding along R}}_{\text{1}}\right)$

$\begin{array}{ll}\Rightarrow & x\left[y\right(1+z)+z]+0+1\left(yz\right)=0\\ \Rightarrow & x(y+yz+z)+yz=0\\ \Rightarrow & xy+xyz+xz+yz=0\\ \Rightarrow & \frac{xy}{xyz}+\frac{xyz}{xyz}+\frac{xz}{xyz}+\frac{yz}{xyz}=0\end{array}$

                                                                $\left[\text{on dividing (}xyz\text{) from both sides}\right]$

$\Rightarrow \frac{1}{x}+\frac{1}{y}+\frac{1}{z}+1=0$

$\Rightarrow \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=-1$

$\therefore x^{-1}+y^{-1}+z^{-1}=-1$